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Lipman Bers
Lipman Bers ( Latvian: ''Lipmans Berss''; May 22, 1914 – October 29, 1993) was a Latvian-American mathematician, born in Riga, who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups. He was also known for his work in human rights activism.. Biography Bers was born in Riga, then under the rule of the Russian Czars, and spent several years as a child in Saint Petersburg; his family returned to Riga in approximately 1919, by which time it was part of independent Latvia. In Riga, his mother was the principal of a Jewish elementary school, and his father became the principal of a Jewish high school, both of which Bers attended, with an interlude in Berlin while his mother, by then separated from his father, attended the Berlin Psychoanalytic Institute. After high school, Bers studied at the University of Zurich for a year, but had to return to Riga again because of the difficulty of transferring money from Latvia in the international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riga
Riga (; lv, Rīga , liv, Rīgõ) is the capital and largest city of Latvia and is home to 605,802 inhabitants which is a third of Latvia's population. The city lies on the Gulf of Riga at the mouth of the Daugava river where it meets the Baltic Sea. Riga's territory covers and lies above sea level, on a flat and sandy plain. Riga was founded in 1201 and is a former Hanseatic League member. Riga's historical centre is a UNESCO World Heritage Site, noted for its Art Nouveau/Jugendstil architecture and 19th century wooden architecture. Riga was the European Capital of Culture in 2014, along with Umeå in Sweden. Riga hosted the 2006 NATO Summit, the Eurovision Song Contest 2003, the 2006 IIHF Men's World Ice Hockey Championships, 2013 World Women's Curling Championship and the 2021 IIHF World Championship. It is home to the European Union's office of European Regulators for Electronic Communications (BEREC). In 2017, it was named the European Region of Gastrono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Nagel
Alexander Joseph Nagel (born 13 September 1945 in New York City) is an American mathematician, specializing in harmonic analysis, functions of several complex variables, and linear partial differential equations. Biography He received in 1966 from Harvard University his bachelor's degree and in 1971 from Columbia University his PhD under the supervision of Lipman Bers with thesis ''Sheaves of Holomorphic Functions with Boundary Conditions and Sheaf Cohomology in Banach Algebras''. At the University of Wisconsin, Madison, Nagel was from 1970 to 1972 an instructor, from 1972 to 1974 an assistant professor, from 1974 to 1977 an associate professor, and from 1977 to 2012 a full professor, retiring in December 2012 as professor emeritus. He was chair of the mathematics department in 1991–1993 and in 2011–2012, and Associate Dean for Natural Sciences in the College of Letters and Science in 1993–1998. He was a Guggenheim Fellow for the academic year 1987–1988. He shared with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Teichmüller Space
In mathematical complex analysis, universal Teichmüller space ''T''(1) is a Teichmüller space containing the Teichmüller space ''T''(''G'') of every Fuchsian group ''G''. It was introduced by as the set of boundary values of quasiconformal maps of the upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ... that fix 0, 1, and ∞. References * * * * * {{DEFAULTSORT:Universal Teichmuller space Riemann surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneous Uniformization Theorem
In mathematics, the simultaneous uniformization theorem, proved by , states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus using a quasi-Fuchsian group of the first kind. The quasi-Fuchsian group is essentially uniquely determined by the two Riemann surfaces, so the space of marked quasi-Fuchsian group of the first kind of some fixed genus ''g'' can be identified with the product of two copies of Teichmüller space of the same genus. References *{{Citation , last1=Bers , first1=Lipman , authorlink=Lipman Bers, title=Simultaneous uniformization , doi=10.1090/S0002-9904-1960-10413-2 , mr=0111834 , year=1960 , journal=Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ... , issn=0002 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoanalytic Function
In mathematics, pseudoanalytic functions are functions introduced by that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations. Definitions Let z=x+iy and let \sigma(x,y)=\sigma(z) be a real-valued function defined in a bounded domain D. If \sigma>0 and \sigma_x and \sigma_y are Hölder continuous, then \sigma is admissible in D. Further, given a Riemann surface F, if \sigma is admissible for some neighborhood at each point of F, \sigma is admissible on F. The complex-valued function f(z)=u(x,y)+iv(x,y) is pseudoanalytic with respect to an admissible \sigma at the point z_0 if all partial derivatives of u and v exist and satisfy the following conditions: :u_x=\sigma(x,y)v_y, \quad u_y=-\sigma(x,y)v_x If f is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain. Similarities to analytic functions * If f(z) is not the constant 0, then the zeroes of f are all isolated. * Therefore, any analytic continuat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Riemann Mapping Theorem
In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and solutions of the Beltrami equation. The result was prefigured by earlier results of Charles Morrey from 1938 on quasi-linear elliptic partial differential equations. The theorem of Ahlfors and Bers states that if μ is a bounded measurable function on C with \, \mu\, _\infty < 1, then there is a unique solution ''f'' of the Beltrami equation : for which ''f'' is a quasiconformal homeomorphism of C fixing the points 0, 1 and ∞. A similar result is true with C replaced by the unit disk
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Density Theorem For Kleinian Groups
In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by and , states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. History suggested the Bers density conjecture, that singly degenerate Kleinian surface groups are on the boundary of a Bers slice. This was proved by for Kleinian surface groups with no parabolic elements. A more general version of Bers's conjecture due to Sullivan and Thurston in the late 1970s and early 1980s states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups. proved this for freely indecomposable Kleinian groups without parabolic elements. The density conjecture was finally proved using the tameness theorem and the ending lamination theorem In hyperbolic geometry, the ending lamination theorem, originally conjectured by , states that hyperb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bers Slice
In the mathematical theory of Kleinian groups, Bers slices and Maskit slices, named after Lipman Bers and Bernard Maskit, are certain slices through the moduli space of Kleinian groups. Bers slices For a quasi-Fuchsian group, the limit set is a Jordan curve whose complement has two components. The quotient of each of these components by the groups is a Riemann surface, so we get a map from marked quasi-Fuchsian groups to pairs of Riemann surfaces, and hence to a product of two copies of Teichmüller space. A Bers slice is a subset of the moduli space of quasi-Fuchsian groups for which one of the two components of this map is a constant function to a single point in its copy of Teichmüller space. The Bers slice gives an embedding of Teichmüller space into the moduli space of quasi-Fuchsian groups, called the Bers embedding, and the closure of its image is a compactification of Teichmüller space called the Bers compactification. Maskit slices A Maskit slice is similar to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ahlfors Finiteness Theorem
In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by , apart from a gap that was filled by . The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed. Bers area inequality The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by . It states that if Γ is a non-elementary finitely-generated Kleinian group with ''N'' generators and with region of discontinuity Ω, then :Area(Ω/Γ) ≤ with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then :Area(Ω/Γ) ≤ 2Area(Ω1/Γ) with equality only for Fuchsian group In math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teichmüller Space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space T(S) may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6g-6 for a surface of genus g \ge 2. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raymond O
Raymond is a male given name. It was borrowed into English from French (older French spellings were Reimund and Raimund, whereas the modern English and French spellings are identical). It originated as the Germanic ᚱᚨᚷᛁᚾᛗᚢᚾᛞ (''Raginmund'') or ᚱᛖᚷᛁᚾᛗᚢᚾᛞ (''Reginmund''). ''Ragin'' ( Gothic) and ''regin'' (Old German) meant "counsel". The Old High German ''mund'' originally meant "hand", but came to mean "protection". This etymology suggests that the name originated in the Early Middle Ages, possibly from Latin. Alternatively, the name can also be derived from Germanic Hraidmund, the first element being ''Hraid'', possibly meaning "fame" (compare ''Hrod'', found in names such as Robert, Roderick, Rudolph, Roland, Rodney and Roger) and ''mund'' meaning "protector". Despite the German and French origins of the English name, some of its early uses in English documents appear in Latinized form. As a surname, its first recorded appearance in B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tilla Weinstein
Tilla Weinstein (1934–2002, née Savanuck, also published as Tilla Klotz and Tilla K. Milnor) was an American mathematician known for her mentorship of younger women in mathematics. Her research concerned differential geometry, including conformal structures, harmonic maps, and Lorentz surfaces. She taught for many years at Rutgers University, where she headed the mathematics department in the Douglass Residential College. Early life and education Weinstein was born as Tilla Savanuck, in 1934. Her father was a Russian immigrant and lawyer in New York City; her mother was a legal secretary. She began her undergraduate studies in 1951 as an English major at the University of Michigan, in part to get away from her parents' rocky marriage and to live near relatives in Detroit. There, her courses included calculus from Hans Samelson and a course in the foundations of mathematics from Raymond Louis Wilder. After her first year in Michigan, she became engaged to an English student ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
